Topological analysis of pseudo-Euclidean Euler top for special values of the parameters

M. K. Altuev   ; V. A. Kibkalo

Sbornik. Mathematics, Tome 214 (2023) no. 3, pp. 334-348

Résumé

An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations. Bibliography: 23 titles.

Détail

Keywords: integrable system, rigid body dynamics, pseudo-Euclidean space, topological invariant, singularity.
Mots-clés : Liouville foliation

Bibliographie

[1] S. Smale, “Topology and mechanics. 1”, Invent. Math., 10:4 (1970), 305–331 | DOI | MR | Zbl

[2] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Udmurtian University Publishing House, Izhevsk, 1999, 444 pp., 447 pp. ; English transl., Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | MR | Zbl | DOI | MR | Zbl

[3] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Dokl. Akad. Nauk SSSR, 287:5 (1986), 1071–1075 ; English transl. in Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[4] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Izv. Akad. Nauk SSSR Ser. Mat., 50:6 (1986), 1276–1307 ; English transl. in Math. USSR-Izv., 29:3 (1987), 629–658 | MR | Zbl | DOI

[5] A. T. Fomenko and Kh. Tsishang (H. Zieschang), “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Izv. Akad. Nauk SSSR Ser. Mat., 54:3 (1990), 546–575 ; English transl. in Math. USSR-Izv., 36:3 (1991), 567–596 | MR | Zbl | DOI

[6] A. V. Bolsinov, S. V. Matveev and A. T. Fomenko, “Topological classification of integrable Hamiltonian systems with two degrees of freedom. List of systems of small complexity”, Uspekhi Mat. Nauk, 45:2(272) (1990), 49–77 ; English transl. in Russian Math. Surveys, 45:2 (1990), 59–94 | MR | Zbl | DOI

[7] A. T. Fomenko and S. V. Matveev, Algorithmic and computer methods for three-manifolds, Moscow University Publishing House, Moscow, 1991, 303 pp. ; English transl., Math. Appl., 425, Kluwer Acad. Publ., Dordrecht, 1997, xii+334 pp. | MR | Zbl | DOI | MR | Zbl

[8] S. S. Nikolaenko, “Topological classification of the Goryachev integrable systems in the rigid body dynamics: non-compact case”, Lobachevskii J. Math., 38:6 (2017), 1050–1060 | DOI | MR | Zbl

[9] S. S. Nikolaenko, “Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds”, Mat. Sb., 211:8 (2020), 68–101 ; English transl. in Sb. Math., 211:8 (2020), 1127–1158 | DOI | MR | Zbl | DOI

[10] V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Ross. Akad. Nauk Ser. Mat., 81:4 (2017), 20–67 ; English transl. in Izv. Math., 81:4 (2017), 688–733 | DOI | MR | Zbl | DOI

[11] E. A. Kudryavtseva and T. A. Lepskii, “Topology of foliations and Liouville's theorem for integrable system with incomplete flows”, Tr. Semin. Vektor. Tenzor. Anal., no. 27, 2011, 106–149 (Russian)

[12] D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$”, Mat. Sb., 202:5 (2011), 127–160 ; English transl. in Sb. Math., 202:5 (2011), 749–781 | DOI | MR | DOI | Zbl

[13] D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{so}(3,1)$”, Mat. Sb., 205:8 (2014), 41–66 ; English transl. in Sb. Math., 205:8 (2014), 1107–1132 | DOI | MR | Zbl | DOI

[14] E. A. Kudryavtseva, “An analogue of the Liouville theorem for integrable Hamiltonian systems with incomplete flows”, Dokl. Ross. Akad. Nauk, 445:4 (2012), 383–385 ; English transl. in Dokl. Math., 86:1 (2012), 527–529 | MR | Zbl | DOI

[15] K. R. Aleshkin, “The topology of integrable systems with incomplete fields”, Mat. Sb., 205:9 (2014), 49–64 ; English transl. in Sb. Math., 205:9 (2014), 1264–1278 | DOI | MR | Zbl | DOI

[16] D. A. Fedoseev and A. T. Fomenko, “Noncompact bifurcations of integrable dynamic systems”, Fundam. Prikl. Mat., 21:6 (2016), 217–243 ; English transl. in J. Math. Sci. (N.Y.), 248:6 (2020), 810–827 | MR | Zbl | DOI

[17] E. A. Kudryavtseva and D. A. Fedoseev, “Mechanical systems with closed orbits on manifolds of revolution”, Mat. Sb., 206:5 (2015), 107–126 ; English transl. in Sb. Math., 206:5 (2015), 718–737 | DOI | MR | Zbl | DOI

[18] O. A. Zagryadskii, E. A. Kudryavtseva and D. A. Fedoseev, “A generalization of Bertrand's theorem to surfaces of revolution”, Mat. Sb., 203:8 (2012), 39–78 ; English transl. in Sb. Math., 203:8 (2012), 1112–1150 | DOI | MR | Zbl | DOI

[19] S. S. Nikolaenko, “Topological classification of noncompact 3-atoms with circle action”, Chebyshevskii Sb., 22:5 (2021), 185–197 (Russian) | DOI | MR

[20] V. A. Kibkalo, “Noncompactness property of fibers and singularities of non-Euclidean Kovalevskaya system on pencil of Lie algebras”, Vestn. Moskov. Univ. Ser. 1 Mat. Mekh., 6 (2020), 56–59 ; English transl. in Moscow Univ. Math. Bull., 75:6 (2020), 263–267 | MR | Zbl | DOI

[21] C. V. Sokolov, “Kovalevskaya's integrable case in a non-Euclidean space: separation of variables”, Tr. Mosk. Aviatsionn. Inst., 100 (2018), 4, 13 pp. (Russian)

[22] A. V. Borisov and I. S. Mamaev, “Rigid body dynamics in non-Euclidean spaces”, Russ. J. Math. Phys., 23:4 (2016), 431–454 | DOI | MR | Zbl

[23] Classical dynamics in non-Euclidean spaces, eds. A. V. Borisov and I. S. Mamaev, Institute for Computer Studies, Moscow–Izhevsk, 2004, 348 pp. (Russian)